\1cw  Verze 2.10
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\"1 0.  C O M B I N A T O R I C S   O F   N A T U R A L \,
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\"  V E C T O R S  \,
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\410.1 The polynomial coefficient\,
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     A partition of the number m into n parts is an n-dimensional 
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vector \ \4m \1which elements \ are \ ordered \ in the \ decreasing order, 
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m \9> \1m   . From this \ vector all other vectors on \ the given orbit 
\-\2\ j  \ j+1
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can be generated, if its elements are permuted by the unit permu-\A
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tation matrices \ acting on the \ partition vector from \ the right. 
\-
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These vectors correspond to the scalar products of naive matrices 
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\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ T\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ T
\4N \1with the \ unit vector rows \ \4J\1ö or to \ the quadratic forms \ \4N\1ö\4N\1, 
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\+\2\ \ \ \ \ \ \ \ \ T\ \ \ \ \ T\ T
\1because \4J N \1= \4J N N\1. There are n! permutation matrices, but not so 
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many permuted vector columns, if \ some elements of the vector row 
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are not distinguishable. Vectors having \ equal length m  point to 
\-\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ k
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the sphere \ and if rotate,they \ remain on it. Therefore, \ they are 
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undistinguishable at permutations. If all parts are equal, then a-\A
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ll permutations have no effect on the partition vector.\,
\-
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     We divide vector \ elements into two groups, in \ one all zero 
\-\2\ \ \ \   \/
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elements) will be , it is n  elements, and in other group all re-\A
\-\2\ \ \ \ \ \ \  \ \ \ \ \ \ \ \ \ \          0
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maining elements (n-n ). The number of possible permutations will 
\-\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0
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\+\0
\1be \ reduced from \ the factorial \ n! on \ the binomial \ coefficient 
\-\0\/
\+\1
\+\0( \1n \0)
\1     , or n!/n\ !(n-n )!. In the next step \ we single out vectors 
\-\09 \1n\20\00\ \ \ \ \ \ \ \ \ \20\ \ \ \ \ 0\/
\+\1
\+\0
\1with the length 1, their number \ is n , all other vectors will be 
\-\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1
\+\1
\+\0
\1counted by the term \ (n-n -n ), and corresponding permutations by 
\-\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0\ \ 1
\+\1
\+\0
\1the \ binomial coefficient \ (n-n )!/n !(n-n -n )!. So \ we proceed, 
\-\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0\ \ \ \ 1\ \ \ \ \ 0\ \ 1
\+\1
\+\0
\1till all possible values of m  will be exhausted. If some n  = 0, 
\-\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ k\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  \ \ \ \ \ \ \ \ k
\+\1
\+\0
\1then conveniently 0! = 1 and the corresponding term is ineffecti-\A
\-\2\/
\+\1
\+\0
\1ve. At the end, we obtain a product of binomial coefficients:\,
\-\2
\+\1
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \   \ \ \ \ \ m-1
\1n!         (n-n\ )!       (n-n\ -n\ )!         (n-\7S\ \ \ \1n\ )!\,
\-\2\ \ \ \ \ \ \ \ \ \   \ \ \ 0\ \ \ \ \ \ \ \ \ \ \ \ \ 0\ \ 1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ k=0\ k
\+\8---------\ \ -------------\ ----------------\ \1..\8-----------
\+\1
n\ !(n-n\ )!\ n\ !(n-n\ -n\ )! n\ !(n-n\ -n\ -n\ )!\ \   n\ !0!\,
\-\2\ 0\ \ \ \ \ 0\ \ \ \ 1\ \ \ \ \ 0\ \ 1\ \ \ \ 2\ \ \ \ \ 0\ \ 1\ \ 2\ \ \ \   \ m
\+\1
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   Equal factorials \ appear consecutively as \ dividends and divi-\A
\-
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sors. When they cancel, there remains the polynomial coefficient \,
\-
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\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ n!/\7P\ \ \ \1n\ !              (10.1)\,
\-\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ k\9>\20\ k
\+\1
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   We will call it the polynomial coefficient for n permutations,
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because it is obtained by \ permuting n columns. Later we will con-\A
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struct another polynomial coefficient for permutations of rows of 
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naive matrices.\,
\-
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   We limited the index k by the lower limit 0. Actually,the coef-\A
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ficient could be used also for vectors with negative elements, n  
\-\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ k
\+\1
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are always positive numbers, but we \ count by the polynomial coef-\A
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ficient (10.1) points on the partition orbits of the positive co-\A
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ne of the n dimensional space.\,
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     Please, note the importance of this step. We know the vector 
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\4m \1exactly, but we replace it \ by the corresponding partition. All 
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points on \ the given orbit \ are considered to \ be \4equivalent\1. The 
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replacing the vector \4m \1by the partition is a logical abstraction. 
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We can proceed further, the \ partition is compared with an analy-\A
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tical function and the orbit \ is described by a density distribu-\A
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tion.\,
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\410.2 Simplex sums of polynomial coefficients\,
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     Now it is \ possible to apply again partition \ schemes and to 
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study sums of polynomial coefficients on all orbits of plane sim-\A
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plices, it means all \ natural n-dimensional vectors with constant 
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sums m.\,
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   The overall sum is known \ in combinatorics as the distribution 
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of m \ undistinguishable things into n \ boxes. It is counted \ by a 
\-
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binomial coefficient\,
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\+\0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (\1m+n-1\0)\ \ (\1m+n-1\0)
\7\ \ \ \ \ \ \ \ \ \ \ S\ \ \ \1n!/\7P\1n\ ! = \ \ \ \ \ \  = \ \ \ \ \ \ \         (10.2)\,
\-\2\ \ \ \ \ \ \ \ \ \ \ \ k\9>\20\ \ \ \ \ k\ \ \ \ \09  \1m  \00\ \ 9 \1n-1 \00
\+\1
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     Both binomial \ coefficients are only different \ forms of one 
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coefficient. Most \ easily this binomial \ coefficient is obtained, 
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if we follow \ all possibilities, how to divide \ m things in a row 
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by (n-1) bars representing dividing walls of compartments. We ha-\A
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ve (m+n-1) objects of two kinds and the result is simply given by 
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a binomial coefficient. \ Who is not satisfied \ with this explana-\A
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tion, can prove (10.2) by the full induction. We tried it at sim-\A
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ple cases and it functioned well. Thus we suppose that it is true 
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for all n dimensional vectors with \ (m-1) elements and to all (n-
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1) dimensional vectors with m elements. \ We use them for counting 
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points with sums m in n-dimensions. These we divide into two dis-\A
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\1tinct sets. In one all points \ having the last element 0 will be. 
\-\0
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\1Clearly, they all are in \ (n-1)-dimensional subspace and they are 
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\+\0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (\1m+n-2\0)
\1counted by the binomial coefficient \        .. In the second sub-\A
\-\0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 9  \1m  \00
\+\1
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\1set vectors having the last \ element at least 1 are counted. They 
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\1are obtained from partitions of (m-1) things into exactly n parts 
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\1by adding \ 1 to the first \ element.This addition does \ not change 
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\1the corresponding number of points. The result is formed by a sum 
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of 2 binomial coefficients\,
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(m+n-2)!\ \ \ \ (m+n-2)!    \ \  (m+n-2)![(n-1)+m]\ \ \ \0(m\1m+n-1\0)\,
\-\8--------\ \1+\ \8-------------\ \1= \8------------------\ \1=\ \ \ \ \ \ \ \ \ \ (10.3)
\+m!(n-2)!    (m-1)!(n-1)!   m!(n-1)!          \ \ \09  \1m   \00
\+\1
\,
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We have told that we will not be interested in vectors with nega-\A
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tive values, but it is \ instructive to show results according the 
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lover limit of \ the value r which appears \ as the parameter (1-r) 
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at \ n \ in \ the \ binomial \ coefficients. \ It \ can be considered as 
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differentiating of the simplex\,
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\+\8\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\1Lower limit             \   -1          0         1          2\,
\-\8-----------------------k-----------------------------------------
\+\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\+\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \0(\1m+2n-1\0)    (\1m+n-1\0)   (\1m-1\0)     (\1m-n-1\0)
\1Points on the simplex\ \ \8p\,
\-\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \09 \1n-1  \00    9 \1n-1 \00   9\1n-1\00     9 \1n-1 \00
\+\1
\+\0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (\1m+3-1\0)
\1   The binomial coefficients          are \ known as triangle num-\A
\-\0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 9  \1m  \00\/
\+\1
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\1bers. They count points of 3 dimensional planes, triangles.\,
\-
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\410.3 Differences of normalized simplices\,
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   We have counted points of the plane simplices directly, now we 
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apply partition schemes and insert into them polynomial coeffici-\A
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ents, similarly as cycle indices \ in Chapter 8. We limit ourselves 
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on cases when m=n. As an example, we give the scheme for m=n=6:\,
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\+\8\ \ \ \ \ p
\1n    \8p  \11    2    3    4    5    6\,
\-
\+\8-----k-----------------------------
\+\ \ \ \ \ p
\1m=6  \8p  \16\,
\-\8\ \ \ \ \ p
\+\ \ \ \ \ p
\+\ \ \ \ \ p
\1  5  \8p      \130\,
\-\8\ \ \ \ \ p
\+\ \ \ \ \ p
\+\ \ \ \ \ p
\1  4  \8p      \130   60\,
\-\8\ \ \ \ \ p
\+\ \ \ \ \ p
\+\ \ \ \ \ p
\1  3  \8p      \115  120   60\,
\-\8\ \ \ \ \ p
\+\ \ \ \ \ p
\+\ \ \ \ \ p
\1  2  \8p           \120   90   30\,
\-\8\ \ \ \ \ p
\+\ \ \ \ \ p
\+\ \ \ \ \ p
\1  1  \                            1\,
\-\8-----k-----------------------------
\+\ \ \ \ \ p
\+\ \ \ \ \ p
\1Sums \8p \16    75  200  150   30    1\,
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   In the \ first column are \ counted vertices of \ the simplex, in 
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the second column points on \ 2-dimensional edges, in the 3. column 
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points of \ its 3-dimensional sides \ and only the \ last point lies 
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inside of \ the 6 dimensional plane, \ all other points \ are on its 
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borders. This is \ a rather surprising property of \ the high dimen-\A
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sional spaces. But we \ can not forget, that usually \ m>>n and then 
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there are more points inside than on the border.\,
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   Column sums \ of consecutive normalized plane \ simplices can be 
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arranged \ into Table \ 10.1, which \ rows are \ known as \ the Van der 
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Monde identity:\,
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\4Table 10.1 Van der Monde identity\,
\-\1
\+\8\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\4n   \8p  \41    2    3    4    5    6 \8p   \7S\,
\-\1
\+\8----k-----------------------------k----
\+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\1m=1 \8p  \11                          \8p   \11\,
\-\8\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\1  2 \8p  \12    1                     \8p   \13\,
\-\8\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\1  3 \8p  \13    6    1                \8p  \110 \,
\-\8\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\1  4 \8p  \14   18   12    1           \8p  \135\,
\-\8\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\1  5 \8p  \15   40   60   20    1      \8p \1126\,
\-\8\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\1  6 \8p  \16   75  200  150   30    1 \8p \1462\,
\-
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   The \ elements in \ each  \ row \ can \ be written \ as products \ of \,
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2 binomial coefficients, e. g. \ 75 = (6!/4!2!)*(5!/4!1!). We have 
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a special case of the identity\,
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\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ m-k\0( \1m \0)( \1m-k\0)\ \ \ \ (\1m+k\0)
\7\ \ \ \ \ \ \ \ \ \ \ \ S\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \1=\ \ \ \ \ \ \             (10.4)\,
\-\2\ \ \ \ \ \ \ \ \ \ \ \ \ i=0\09\1k+i\009  \1i \00\ \ \ \ 9\ \1m\ \00
\+\1
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The sum of products of 2 \ binomial coefficients can be written as 
\-
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a formal power of a binomial\,
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\+\0\ \ \ \ \ \ \ \ \ &( \1m \0)\ \ \ \ \ *\1n\ \ \ \0(m\1m+n \0)
\1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ +  1\ \ \  = \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (10.4)\,
\-\0\ \ \ \ \ \ \ \ \ 79 \1i \00\ \ \ \ \ 8\ \ \ \ 9  \1m  \00
\+\1
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ n/2\0(\1n\0)\12   \0(\12n\0)
\1Its special case is the Waltis identity for m=n: \7S\ \ \ \ \ \ \ \ \1=\,
\-\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ i=0\09\1i\00    9 \1n\00
\+\1
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We interpret it as the simplex \ in which the first vector is roo-\A
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ted and only (n-1) other vectors are permuted. For example:\,
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\+\8\ \ \ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ p
\1Orbits \  4000 \  3100   1300   2200 \  2110   1210 \  1111\   \7S\,
\-\8-------k------k--------------------k-------------k-----k----
\+\ \ \ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ p
\+\ \ \ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ p
\1Points \8p \11    \8p \13      3      3    \8p \13      6    \8p \11\ \  \8p \120\,
\-\8\ \ \ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ p
\+\ \ \ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ p
\+\ \ \ \ \ \ \ p\ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ p
\1Counts \8p \11    \8p        \19           \8p    \19        \8p \11   \8p \120\,
\-
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\410.4 Difference according to unit elements\,
\-\1
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   When we \ arrange the partition scheme \ according to the number 
\-\/
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of unit \ vectors n , we obtain a\ difference of the \ plane simplex.\,
\-\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1 
\+\1
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E.g. for m=n=5 \,
\-\2
\+\1
\+\8\ \ \ \ p
\1n   \8p  \10   1   2   3   4   5\,
\-\2\ 1
\+\8----k------------------------
\+\ \ \ \ p
\1m=5 \8p  \15\,
\-\8\ \ \ \ p
\+\ \ \ \ p
\+\ \ \ \ p
\1  4 \8p     \120 \,
\-\8\ \ \ \ p
\+\ \ \ \ p
\+\ \ \ \ p
\1  3 \8p \120      30\,
\-\8\ \ \ \ p
\+\ \ \ \ p
\+\ \ \ \ p
\1  2 \8p     \130      20\,
\-\8\ \ \ \ p
\+\ \ \ \ p
\+\ \ \ \ p
\1  1 \8p                      \11\,
\-
\+\8----k-----------------------
\+\ \ \ \ p
\7  S \8p \125  50  30  20   0   1
\-
\+
\+
     The resulting column sums of polynomial coefficients are ta-\A
\-\/
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\+
bulated in Table 10.2:\,
\-
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\4Table 10.2 Unit elements difference\,
\-\1
\+
\+\8\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\1n   \8p \10    1    2    3    4    5    6 \8p  \7S\,
\-\2\ 1
\+\8----k---------------------------------k---
\+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\1m=0 \8p \11                               \8p  \11\,
\-\8\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\1  1 \8p \10    1                          \8p  \11\,
\-\8\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\1  2 \8p \12    0    1                     \8p  \13\,
\-\8\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\1  3 \8p \13    6    0    1                \8p \110\,
\-\8\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\1  4 \8p\110   12   12    0    1           \8p \135\,
\-\8\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\1  5 \8p\125   50   30   20    0    1      \8p\1126\,
\-\8\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\1  6 \8p\171  150  150   60   30    0    1 \8p\1462 \,
\-
\+
\+
   The numbers \ b   are formed by vectors \ without any unit \ ele-\A
\-\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ i0   \/
\+\1
\+
ments. They can be called \ subplane numbers, because they generate 
\-
\+
\+
the number of \ points of the normal plane \ simplex by multiplying 
\-
\+
\+
with binomial coefficients:\,
\-
\+
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ m\ \ \ \0(\1m+n-1\0)
\1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (b\  + 1)\  = \ \ \ \ \ \ \       (10.5)\,
\-\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ k\ \ \ \ \ \ \ \ \ \09  \1m  \00
\+\1
\+
     They are (n-k) dimensional vectors without unit elements but 
\-\/
\+
\+
with zero elements. Their (n-k) elements are combined with k unit 
\-
\+
\+
elements. When \ m \9$ \1n, such relations \ are more complicated. Cor-\A
\-
\+
\+
responding subplane numbers are \ obtained by corresponding calcu-\A
\-
\+
\+
lations for partitions without units. \ The beginning of the table 
\-\/
\+
\+
is\,
\-
\+\8\ \ \ \ p
\+\ \ \ \ p
\1n   \8p  \10   1   2   3   4   5   6\,
\-
\+\8----k---------------------------
\+\ \ \ \ p
\1m=0 \8p  \11   1   1   1   1   1   1\,
\-\8\ \ \ \ p
\+\ \ \ \ p
\+\ \ \ \ p
\1  1 \8p  \10   0   0   0   0   0   0\,
\-\8\ \ \ \ p
\+\ \ \ \ p
\+\ \ \ \ p
\1  2 \8p  \10   1   2   3   4   5   6\,
\-\8\ \ \ \ p
\+\ \ \ \ p
\+\ \ \ \ p
\1  3 \8p  \10   1   2   3   4   5   6\,
\-\8\ \ \ \ p
\+\ \ \ \ p
\+\ \ \ \ p
\1  4 \8p  \10   1   3   6  10  15  21\,
\-\8\ \ \ \ p
\+\ \ \ \ p
\+\ \ \ \ p
\1  5 \8p  \10   1   4   9  16  25  36\,
\-\8\ \ \ \ p
\+\ \ \ \ p
\+\ \ \ \ p
\1  6 \8p  \10   1   5  13  26  45  71\,
\-
\+
\+
     Its values b(i,j)   for small m are:\,
\-\2
\+\1
\+\0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (\1n\0)\          (\1n\0)  
\1b(0,n) = 1; B(1,n) = 0; b(2,n) =    ; b(3,n) =   ; \,
\-\0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 9\11\00\          9\11\00  
\+\1
\+\0         (\1n\0)(\1n\0)   (\1n+1\0)           (\1n\0)    (\1n\0)    \22  
\1b(4,n) =        =      ; b(5,n) =     + 2    = n ; \,
\-\0         9\11\009\12\00   9 \12 \00           9\11\00    9\12\00     
\+\1
\+\0         (\1n\0)    (\1n\0)\  \ (\1n\0)
\1b(6,n) =     + 3    +\ \ \ \ .\,
\-\0         9\11\00    9\12\00  \ 9\13\00
\+\1
\+
     The subplane numbers appear here on the diagonal. An example 
\-
\+
\+
of their application for m = 4, n = 6:\,
\-
\+
\+\0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (\19\0)
\121 + 6*5 + \ 15*4 + 20*0 + 15*1 = 126 \ = ööö. Vectors without unit 
\-\0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 9\14\00
\+\1
\+
elements are combined with unit vectors.\,
\-
\+\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \   
\+
\410.5 Differences according to one element\,
\-\1
\+
\+
     In partition schemes the points are counted in spherical or-\A
\-\/
\+
\+
bits. We can differentiate the plane according to only one speci-\A
\-\/
\+
\+
fic vector x. It can be shown on the 2-dimensional complex:\,
\-
\+
\+\8\ \ \ \ \ \ \ \ p
\1m\       \8p  \10   1   2   3   4   5\,
\-\2\ a
\+\8--------k-----------------------
\+\ \ \ \ \ \ \ \ p
\1Points  \8p \ \1*   *   *   *   *   *\,
\-\8\ \ \ \ \ \ \ \ p
\+\ \ \ \ \ \ \ \ p
\+\ \ \ \ \ \ \ \ p
        p      \1*   *   *   *   *\,
\-\8\ \ \ \ \ \ \ \ p
\+\ \ \ \ \ \ \ \ p
\+\ \ \ \ \ \ \ \ p
        p          \1*   *   *   *\,
\-\8\ \ \ \ \ \ \ \ p
\+\ \ \ \ \ \ \ \ p
\+\ \ \ \ \ \ \ \ p
        p              \1*   *   *\,
\-\8\ \ \ \ \ \ \ \ p
\+\ \ \ \ \ \ \ \ p
\+\ \ \ \ \ \ \ \ p
        p                  \1*   *\,
\-\8\ \ \ \ \ \ \ \ p
\+\ \ \ \ \ \ \ \ p
\+\ \ \ \ \ \ \ \ p
        p                      \1*\,
\-
\+\8--------k-----------------------
\+\ \ \ \ \ \ \ \ p
\1Number  \8p \11   2   3   4   5    6\,
\-
\+
\+
   2-dimensional \ complex forms \ a 3-dimensional \ simplex and its 
\-
\+
\+
points for different values of the vector a are counted by column 
\-
\+
\+
sums. It is similar, as when \ points of the (n-1) dimensional com-\A
\-\/
\+
\+
plex are counted for different values \ of m, m  going from 0 till 
\-\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ k
\+\1
\+\0
\1m. \ \ These \ \ points \ \ are \ \ counted \ \ by \ \ binomial \ coefficients 
\-\0\/
\+\1
\+\0(\1n+k-2\0)
\1       . For example, n=7, m=7: \,
\-\09  \1k  \00
\+\1
\+
m\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \8p\,
\-\1\ k\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0    1    2    3    4    5    6    7
\+\8---------------------k----------------------------------------
\+\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\1Binomial \ coefficient\8p\ \1792\ \ 462 \ 252\ \ 123\ \ \ 56\ \ \ 21\ \ \ \ 6\ \ \ \ 1\,
\-
\+
\+\0
\1     We obtain the identity\,
\-\0
\+\1
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ m\ \0(\1n+k-2\0)\ \ \ (\1n+m-1\0)
\7\ \ \ \ \ \ \ \ \ \ \ \ \ \ S\ \ \ \ \ \ \ \ \ \  \1= \ \ \ \ \ \ \           (10.6)\,
\-\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ k=0\09  \1k  \00\ \ \ 9  \1m  \00
\+\1
\+
     Now, we introduce another difference.\,
\-
\+
\+
\410.6 Difference \7D\1(n) \4of plane simplices\,
\-\1
\+
\+
     Till now, \ zero elements were permuted \ with other elements. 
\-\/
\+
\+
We exclude them \ and we will count only \ existing vectors and not 
\-
\+
\+
virtual ones. \ It means that we \ will count k-dimensional vectors 
\-
\+
\+
with constant sums m. If \ we draw a tetrahedron, then the counted 
\-
\+
\+
set of points \ is formed by one vertex, one \ edge without the se-\A
\-
\+
\+
cond vertex, the \ inside of one side and \ by the four dimensional 
\-
\+
\+
core. In \ combinatorics these vectors are \ known as compositions. 
\-\/
\+
\+
There can be arranged onto partition schemes. For m=5 we get:\,
\-
\+\8\ p\ \ p\ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ p
\+\ p\ \ p\ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ p
\1n\8p \11\8p   \12   \8p      \13        \8p          \14           \8p   \15    \8p \7S\,
\-\1
\+\8-k--k-------k---------------k----------------------k--------k----
\+\ p\ \ p\ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ p
 p \15\8p       p               p                      p        p \11\,
\-\8\ p\ \ p\ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ p
\+\ p\ \ p\ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ p
\+\ p\ \ p\ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ p
 p  p \141,14 \8p               p                      p        p \12\,
\-\8\ p\ \ p\ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ p
\+\ p\ \ p\ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ p
\+\ p\ \ p\ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ p
 p  p \132,23 \8p \1311,131,113,  \8p                      p        p \15\,
\-\8\ p\ \ p\ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ p
\+\ p\ \ p\ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ p
\+\ p\ \ p\ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ p
 p  p       p \1221,212,122;  \8p  \12111,1211,1121,1112 \8p        p \17\,
\-\8\ p\ \ p\ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ p
\+\ p\ \ p\ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ p
\+\ p\ \ p\ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ p
 p  p       p               p                      p\111111   \8p \11\,
\-
\+\8-k--k-------k---------------k----------------------k--------k---
\+\ p\ \ p\ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ p
\7S\8p \11\8p  \14    \8p     \16         \8p          \14           \8p  \11     \8p\116\,
\-\8\ p\ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\+\1
\+
     For normal simplices we obtain Table 10.3\,
\-
\+
\+
\4Table 10.3 Binomial coefficients (matrix B)\,
\-\1
\+
\+\8\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\1k   \8p  \11    2    3    4    5    6 \8p  \7S\,
\-\1
\+\8----k-----------------------------k----
\+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\1m=1 \8p  \11                          \8p  \11\,
\-\8\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\1  2 \8p  \11    1                     \8p  \12\,
\-\8\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\1  3 \8p  \11    2    1                \8p  \14\,
\-\8\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\1  4 \8p  \11    3    3    1           \8p  \18\,
\-\8\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\1  5 \8p  \11    4    6    4    1      \8p \116\,
\-\8\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\1  6 \8p  \11    5   10   10    5    1 \8p \132\,
\-
\+
\+
    The table \ has one fault, \ both indices must \ be decreased by 
\-
\+
\+\0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (\1k-1\0)
\1one, to \ obtain \ the \ true \ binomial \ coefficient \ ööööö. \ We had 
\-\0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 9\1m-1\00
\+\1
\+\0
\1difficulties with \ the binomial coefficient \ even before, when it 
\-\0
\+\1
\+\0\ \ \ \ \ \ \ \ \ \ \ \ (\1m+n-1\0)
\1appeared as \ \ \ \ \ \ \ . In this case, it is the Table 10.4:\,
\-\0\ \ \ \ \ \ \ \ \ \ \ \ 9  \1m  \00
\+\1
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ T
\4Table 10.4 Matrix B\ B of binomial coefficients\,
\-\1
\+
\+\8\ \ \ \ \ p
\1n    \   1    2    3    4    5    6\,
\-\8-----k----------------------------
\+\ \ \ \ \ p
\+\ \ \ \ \ p
\1m=0  \8p  \11    1    1    1    1    1\,
\-\8\ \ \ \ \ p
\+\ \ \ \ \ p
\+\ \ \ \ \ p
\1  1  \8p  \11    2    3    4    5    6\,
\-\8\ \ \ \ \ p
\+\ \ \ \ \ p
\+\ \ \ \ \ p
\1  2  \8p  \11    3    6   10   15   21\,
\-\8\ \ \ \ \ p
\+\ \ \ \ \ p
\+\ \ \ \ \ p
\1  3  \8p  \11    4   10   20   35   56\,
\-\8\ \ \ \ \ p
\+\ \ \ \ \ p
\+\ \ \ \ \ p
\1  4  \8p  \11    5   15   35   70  126\,
\-\8\ \ \ \ \ p
\+\ \ \ \ \ p
\+\ \ \ \ \ p
\1  5  \8p  \11    6   21   56  126  252\,
\-
\+
\+
     In \ both preceding \ tables of \ binomial coefficients, \ their 
\-\/
\+
\+
elements were obtained similarly, as \ a sum of two neighbors, the 
\-
\+
\+
left one \ and the upper \ one, except that \ in the Table \ 10.4 the 
\-\/
\+
\+
left element is added only if j \9< \1i.\,
\-
\+
\+
     Recall \ transactions, \ we \ made \ with \ partitions \ and their 
\-\/
\+
\+
counts according the lower allowed \ limit of parts. Here, we have 
\-
\+
\+
a similar shift of values of Tables 10.3 and 10.4, but the opera-\A
\-
\+
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ T
\1tion is made by the matrix of binomial coefficients \4B \1. We permu-\A
\-
\+
\+
te k nonzero elements with (n-k) zero elements and from a part of 
\-
\+
\+
the \ plane simplex \ we obtain \ the whole \ simplex. Therefore this 
\-
\+
\+
part is its difference \7D\1. \ Because there are more differences, it 
\-
\+
\+
is the \ difference according to the \ number of vectors n. \ Now we 
\-\/
\+
\+
can return to Table 10.3. Its elements have the reccurence\,
\-
\+
\+
\ \ \ \ \ \ \ \ \ \ \ \ \ b\ \  = 1; b\ \  = b       + b\ \ \ \ \    (10.7)\,
\-\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ 11\ \ \ \ \ \ \ ij\ \ \ \ i,j-1\ \ \ \ \ i-1,j
\+\1
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  \ \ \ \   \ \ \ \ \ \ \ \ \ m
\1     They are generated by the binomial (1  + 1)  = 2 . We alrea-\A
\-\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \    \ i\/
\+\1
\+
dy formulated the reccurence formula of the Table 10.4 in (10.3). 
\-\2
\+\1
\+
Notice that elements \ of the Table 10.4 are \ sums of all elements 
\-\2
\+\1
\+
of its preceding \ row or column, which is \ the consequence of the 
\-\2\/
\+\1
\+
consecutive applications of (10.3).\,
\-\2
\+\1
\+
     The inverse matrix to the matrix \4B \1is obtained from the for-\A
\-\/
\+
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \   \ \ \ \ \ m
\1mal binomial (1 - 1) = 0. It \ is just the matrix \4B \1which elements 
\-\2\ \ \ \ \ \ \ \ \ \ \ \   \ i\/
\+\1
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ j\1-\2i
\1are multiplied by alternating signs (-1)   .\,
\-
\+
\+
\410.7 Difference \7D\1(m)\,
\-
\+
\+
     When we arranged vector \ compositions in a table, we treated 
\-\/
\+
\+
till now \ only its column \ sums. There are \ also row sums, \ which 
\-
\+
\+
count compositions classified according \ to their greatest vector 
\-\2
\+\1
\+
m . \  The consecutive \ results for \ n=m can \ be arranged into the 
\-\2\ k\/
\+\1
\+
table\,
\-
\+
\+
\4Table 10.5 Composition of vectors with m\  part\,
\-\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ k
\+\8\ \ \ \ \ p
\+\ \ \ \ \ p
\1n    \8p\11   2   3   4   5   6\,
\-
\+\8-----k---------------------
\+\ \ \ \ \ p
\1m\ =1 \8p\11   1   1   1   1   1\,
\-\2\ k\ \ \ \8p
\+\ \ \ \ \ p
\+\ \ \ \ \ p
\1   2 \8p    \11   2   4   7  12\,
\-\8\ \ \ \ \ p
\+\ \ \ \ \ p
\+\ \ \ \ \ p
\1   3 \8p        \11   2   5  11\,
\-\8\ \ \ \ \ p
\+\ \ \ \ \ p
\+\ \ \ \ \ p
\1   4 \8p            \11   2   5\,
\-\8\ \ \ \ \ p
\+\ \ \ \ \ p
\+\ \ \ \ \ p
\1   5 \8p                \11   2\,
\-\8\ \ \ \ \ p
\+\ \ \ \ \ p
\+\ \ \ \ \ p
\1   6 \8p                    \11\,
\-
\+\8-----k----------------------
\+\ \ \ \ \ p
\7S    \8p\11   2   3   8  16  32\,
\-\8\ \ \ \ \ p
\+\1
\+
     The \ elements of \ the Table \ 10.5 are \ related by a somewhat 
\-\/
\+
\+
complicated reccurence\,
\-
\+
\+
c(i,j) -2c(i,j-1) +c(i,j-i-1) = c(i-1,j-1) -2c(i-1,j-1) +c(i-i)\,
\-
\+
\+
with c(1,1) = \ 1 and c(0,x) = 0. It gives \ the uniform first row, 
\-\/
\+
\+
and the upper triangular form of the matrix. \,
\-
\+
\+
\410.7 The second difference - Fibonacci numbers\,
\-\1
\+
\+
     When we \ admit as the smallest \ element 2, we get \ the Table 
\-\/
\+
\+
10.5 \ of points \ of truncated \ plane simplices. \ Its row sums are 
\-
\+
\+
known as \ Fibonacci numbers. In \ a medieval arithmetic book \ they 
\-
\+
\+
appeared as the answer on a number of rabbit pairs in consecutive 
\-\/
\+
\+
litters.\,
\-
\+
\+
\4Table 10.5 Fibonacci numbers\,
\-\1
\+
\+\8\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\1n    \    1   2   3  \  \7S\,
\-\8-----k--------------k--
\+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\1m=2  \8p   \11          \8p \11\,
\-\8\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\1  3  \8p   \11          \8p \11\,
\-\8\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\1  4  \8p   \11   1      \8p \12\,
\-\8\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\1  5  \8p   \11   2      \8p \13\,
\-\8\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\1  6  \8p   \11   3   1  \8p \15\,
\-\8\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\+\ \ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\1  7  \8p   \11   4   3  \8p \18\,
\-
\+
\+
    The vectors \ at m=7 are: \ 7; 52, 25, \ 43, 34; 322, \ 232, 223. \,
\-
\+
\+
Notice, that elements of the Table 10.5 are binomial coefficients 
\-
\+
\+
shifted in each column for \ 2 rows. Fibonacci numbers F  have the 
\-\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ m
\+\1
\+
reccurence F  = F    + F   . The elements \ of the Table 10.5, f   
\-\2\ \ \ \ \ \ \ \ \ \ \ \ m    m-1\ \ \ \ m-2\ \ \ \    \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \     ij  \/
\+\1
\+
are obtained by \ adding 2 to each vector with \ (j-1) nonzero ele-\A
\-\2\/
\+\1\ \  
\+
ments or 1 to the greatest element of the j dimensional vectors\,
\-\2
\+\1
\+
\ \ \ \ \ \ \ \ \ \ \ \ \ f   = 1; f\ \  = f        + f        (10.8)\,
\-\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ 21\ \ \ \ \ \ \ ij\ \ \ \ i-2,j-1    i-1,j 
\+\1
\+
     In each row, all elements \ of both preceding rows are repea-\A
\-\/
\+
\+
ted which gives the reccurence of Fibonacci numbers.\,
\-
\+
\+
     Another \ way, how \ to obtain \ the Fibonacci \ numbers is \ the 
\-\/
\+
\+
count \ of the \ compositions, which \ all elements \ are odd. We get 
\-\/
\+
\+
a scarce Pascal triangle\,
\-
\+\8\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\1n   \   1   2   3   4   5   6  \  \7S\,
\-\8----k-------------------------k--
\+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\1m=1 \8p  \11                      \8p \11\,
\-\8\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\1  2 \8p  \10   1                  \8p \11\,
\-\8\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\1  3 \8p  \11   0   1              \8p \12\,
\-\8\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\1  4 \8p  \10   2   0   1          \8p \13\,
\-\8\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\1  5 \8p  \11   0   3   0   1\ \ \ \ \ \ \8p\ \15\,
\-\8\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\+\ \ \ \ p\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p
\1  6 \8p  \10   3   0   4   0   1  \8p \18\,
\-
\+
\+
     E. g. the last row: 51, 15, 33; 4*(3111); 111111.\,
\-
\+
\+
\410.9 Fibonacci spirals\,
\-\1
\+
\+
     If we draw on two orthogonal axes consecutive Fibonacci num-\A
\-\/
\+
\+
bers, \ hypotenuses connecting \ following points \ of corresponding 
\-\2
\+\1
\+
right triangles are square roots of the squared Fibonacci numbers 
\-\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \         \/
\+\1
\+
F    . This implies the identity\,
\-\2\ 2k+1
\+\1
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2\ \ \ \ \ \ \ 2
\1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Fööööö=öFööööö+öFö        (10.9) \,
\-\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  2k+1    k+1     k
\+\1
\+\2
\1     A similar identity is obtained for \ even numbers as the dif-\A
\-\/
\+
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2\ \ \ \ 2
\1ference of \ two squared Fibonacci numbers \ e.g. F  = F  - F  = 21 
\-\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 8    5 \ \ \ 3
\+\1
\+\2
\1= 25 - 4. This difference can be \ written as a sum of products of 
\-\2\/
\+\1
\+\2
\1Fibonacci numbers.\,
\-\2
\+\1
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2      2\ \ \ \ \ \ 2
\1\ \ \ \ \ \ \ \ \ \ \ F    = F    - F    \2= \1F  + F F        (10.10)\,
\-\2\ \ \ \ \ \ \ \ \ \ \  2k\ \ \ \ \ k+1    k-1\ \ \ \ k\ \ \ \ k k-1
\+\1
\+
     We decompose high Fibonacci \ numbers consecutively and express  
\-
\+
\+
coefficients by lower Fibonacci numbers as: \,
\-
\+
\+
F     = F F   + F F     = F F    + F F    = till (10.9)\,
\-\2 2k+1    2 2k    1 2k-1    3 2k-1   2 2k-2
\+\1
\+
    There appears still another formula\,
\-
\+
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2\ \ \ \ \ \ \ n
\1\ \ \ \ \ \ \ \ \ \ \ \ \ \ F   F    - F  = (-1)\          ( 10.11)\,
\-\2\ \ \ \ \ \ \ \ \ \ \ \ \ \  n+1 n-1    n
\+\1
\+
E.g. for n=5: 3*8 - 25 = -1. \,
\-
\+
\+
\ \ \ It can be formulated in the matrix form\,
\-
\+
\+\8pp\ \ \ \ \ \ \ \ \ \ \ pp\ \ \ pp\ \ \ \ \ \ \ \ pp\1n
\8pp\1F     F\ \ \ \ \8pp\ \ \ pp \11    1\ \8pp\,
\-pp\ \2n+1\ \ \ n\ \ \ \8pp\ \ \ pp\ \ \ \ \ \ \ \ pp
\+pp\ \ \ \ \ \ \ \ \ \ \ pp\ \2= \8pp\ \ \ \ \ \ \ \ pp
\+pp\ \ \ \ \ \ \ \ \ \ \ pp\ \ \ pp\ \ \ \ \ \ \ \ pp
pp\1F     F\ \ \ \ \8pp\ \ \ pp\ \11\ \ \ \ 0\ \8pp\,
\-pp\ \2n\ \ \ \ \ n-1\ \8pp
\+\1
\+
     It leads to two things. The first one are eigenvalues of the 
\-
\+
\+
matrix, see later \ chapters, the second one is \ the zero power of 
\-
\+
\+
this matrix. It is\,
\-
\+\8
\+pp\ \ \ \ \ \ \ \ pp\10\ \ \ \ \ \8pp\ \ \ \ \ \ \ \ \ pp
pp \11    1\ \8pp\ \ \ \ \ \ pp\ \11\ \ \ \ \ 0\ \8pp\,
\-pp\ \ \ \ \ \ \ \ pp\ \ \ \ \ \ pp\ \ \ \ \ \ \ \ \ pp
\+pp\ \ \ \ \ \ \ \ pp\ \ \ \1=\ \ \8pp\ \ \ \ \ \ \ \ \ pp
\+pp\ \ \ \ \ \ \ \ pp\ \ \ \ \ \ pp\ \ \ \ \ \ \ \ \ pp
pp\ \11\ \ \ \ 0\ \8pp\ \ \ \ \ \ pp\ \10\ \ \ \ \ 1\ \8pp\,
\-\1
\+
\+
     On the \ diagonal the values \ Fööö and Fööö \ are. It gives \ a 
\-\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ n+1\ \ \ \ \ \ n-1
\+\1
\+
possibility \ to \ prolongate \ the \ Fibonacci \ numbers to negative 
\-
\+
\+
indices. This queue \ must be: 1, -1, 2, -3, \ 5, -8, ... We obtain 
\-
\+
\+
them again as the sums \ of two consecutive Fibonacci numbers, row 
\-
\+
\+\2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -1
\1sums of \ the elements of \4B   \ \1or as elements of \ their generating 
\-\/
\+
\+
matrix\,
\-
\+
\+
\8pp\ \ \ \ \ \ \ \ pp\1n\,
\-\8pp \10    1\ \8pp
\+pp\ \ \ \ \ \ \ \ pp
\+pp\ \ \ \ \ \ \ \ pp
pp\ \ \ \ \ \ \ \ pp\,
\-pp\ \11\ \ \ -1\ \8pp
\+\1
\+
\,
\-
\+
\+
\,
\-
\=